Factor out the GCF from the polynomial.
45 x Superscript 7 Baseline plus 81 x Superscript 5 Baseline minus 18 x cubed plus 63


To factor out the Greatest Common Factor (GCF) from the polynomial \( 45x^7 + 81x^5 - 18x^3 + 63 \), follow these steps:

1. **Identify the GCF of the coefficients**:
   - The coefficients are 45, 81, 18, and 63.
   - The GCF of these numbers is 9.

2. **Identify the GCF of the variables**:
   - The variable terms are \( x^7 \), \( x^5 \), and \( x^3 \). 
   - The smallest exponent of \( x \) among these terms is 3. 
   - So, the GCF for the variable part is \( x^3 \).

3. **Factor out the GCF**:
   - The GCF of the entire polynomial is \( 9x^3 \).
   - Factor \( 9x^3 \) out of each term in the polynomial:

\[
45x^7 + 81x^5 - 18x^3 + 63 = 9x^3(5x^4 + 9x^2 - 2) + 63 = 9x^3(5x^4 + 9x^2 - 2) + 63 
\]

Since the constant term (63) does not have an \( x \) term, it is not included in the expression with \( x^3 \).

So, the factored form is:

\[
9x^3(5x^4 + 9x^2 - 2) + 63
\]

Would you like more details or have any questions? Here are some related questions to explore:

1. How do you find the GCF of a set of numbers?
2. What if the polynomial had negative exponents or fractional coefficients?
3. Can you factor out a GCF that is a variable without any coefficients?
4. How does factoring out a GCF help in simplifying complex algebraic expressions?
5. What is the process of factoring a polynomial completely after factoring out the GCF?

**Tip:** Always start by finding the GCF of both coefficients and variables to simplify the polynomial as much as possible.

Learn how to factor out the Greatest Common Factor (GCF) from the polynomial 45x^7 + 81x^5 - 18x^3 + 63 with a step-by-step solution. This problem covers key algebraic concepts, including polynomials, GCF, and factoring techniques. Suitable for students in Grades 9-12.

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